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Assembly Theory is an approximation to algorithmic complexity based on LZ compression that does not explain selection or evolution

Published 11 Mar 2024 in cs.IT, math.IT, and q-bio.BM | (2403.06629v6)

Abstract: We prove the full equivalence between Assembly Theory (AT) and Shannon Entropy via a method based upon the principles of statistical compression renamed `assembly index' that belongs to the LZ family of popular compression algorithms (ZIP, GZIP, JPEG). Such popular algorithms have been shown to empirically reproduce the results of AT, results that have also been reported before in successful applications to separating organic from non-organic molecules and in the context of the study of selection and evolution. We show that the assembly index value is equivalent to the size of a minimal context-free grammar. The statistical compressibility of such a method is bounded by Shannon Entropy and other equivalent traditional LZ compression schemes, such as LZ77, LZ78, or LZW. In addition, we demonstrate that AT, and the algorithms supporting its pathway complexity, assembly index, and assembly number, define compression schemes and methods that are subsumed into the theory of algorithmic (Kolmogorov-Solomonoff-Chaitin) complexity. Due to AT's current lack of logical consistency in defining causality for non-stochastic processes and the lack of empirical evidence that it outperforms other complexity measures found in the literature capable of explaining the same phenomena, we conclude that the assembly index and the assembly number do not lead to an explanation or quantification of biases in generative (physical or biological) processes, including those brought about by (abiotic or Darwinian) selection and evolution, that could not have been arrived at using Shannon Entropy or that have not been reported before using classical information theory or algorithmic complexity.

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References (62)
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[7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. 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URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Sharma, A., Czégel, D., Lachmann, M. et al. Assembly theory explains and quantifies selection and evolution. Nature 622, 321–328 (2023). [3] Hazen, R. M. et al. Molecular assembly indices of mineral heteropolyanions: some abiotic molecules are as complex as large biomolecules. Journal of The Royal Society Interface 21, 20230632 (2024). [4] Marshall, S. M., Moore, D. G., Murray, A. R. G., Walker, S. I. & Cronin, L. Formalising the pathways to life using assembly spaces. Entropy 24, 884 (2022). [5] University of Glasgow. Assembly theory unifies physics and biology to explain evolution and complexity. Press Release (2023). [6] Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hazen, R. M. et al. Molecular assembly indices of mineral heteropolyanions: some abiotic molecules are as complex as large biomolecules. Journal of The Royal Society Interface 21, 20230632 (2024). [4] Marshall, S. M., Moore, D. G., Murray, A. R. G., Walker, S. I. & Cronin, L. Formalising the pathways to life using assembly spaces. Entropy 24, 884 (2022). [5] University of Glasgow. Assembly theory unifies physics and biology to explain evolution and complexity. Press Release (2023). [6] Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D. G., Murray, A. R. G., Walker, S. I. & Cronin, L. Formalising the pathways to life using assembly spaces. Entropy 24, 884 (2022). [5] University of Glasgow. Assembly theory unifies physics and biology to explain evolution and complexity. Press Release (2023). [6] Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). University of Glasgow. Assembly theory unifies physics and biology to explain evolution and complexity. Press Release (2023). [6] Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). 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Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. 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Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D. G., Murray, A. R. G., Walker, S. I. & Cronin, L. Formalising the pathways to life using assembly spaces. Entropy 24, 884 (2022). [5] University of Glasgow. Assembly theory unifies physics and biology to explain evolution and complexity. Press Release (2023). [6] Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). University of Glasgow. Assembly theory unifies physics and biology to explain evolution and complexity. Press Release (2023). [6] Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. 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Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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[59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. 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Formalising the pathways to life using assembly spaces. Entropy 24, 884 (2022). [5] University of Glasgow. Assembly theory unifies physics and biology to explain evolution and complexity. Press Release (2023). [6] Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. 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Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. 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[35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). 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Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). University of Glasgow. Assembly theory unifies physics and biology to explain evolution and complexity. Press Release (2023). [6] Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. 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Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). University of Glasgow. Assembly theory unifies physics and biology to explain evolution and complexity. Press Release (2023). [6] Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. 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Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Kiani, N. A. & Zenil, H. Algorithmically probable mutations reproduce aspects of evolution, such as convergence rate, genetic memory and modularity. Royal Society open science 5, 180399 (2018). [7] Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. An algorithmic information calculus for causal discovery and reprogramming systems. iScience 19, 1160–1172 (2019). [8] Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. 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Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. 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Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Chaitin, G. Algorithmic Information Theory 3 edn (Cambridge University Press, 2004). [9] Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Calude, C. S. Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). 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Information and Randomness: An algorithmic perspective 2 edn (Springer-Verlag, 2002). [10] Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). 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[50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) 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Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. 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URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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[59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). 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Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. 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On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. 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IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. 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[27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Li, M. & Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. 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Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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[59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. 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Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. 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Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  10. An Introduction to Kolmogorov Complexity and Its Applications 4 edn. Texts in Computer Science (Springer, Cham, 2019). [11] Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Downey, R. G. & Hirschfeldt, D. R. Algorithmic Randomness and Complexity (Springer New York, New York, NY, 2010). URL http://link.springer.com/10.1007/978-0-387-68441-3. [12] Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. 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[29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Uthamacumaran, A., Abrahão, F. S., Kiani, N. A. & Zenil, H. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. 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Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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[59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. 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The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). 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  12. On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures. arXiv:2210.00901 [cs.IT] (2022). [13] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. 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Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 (1948). [14] Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) 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[50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) 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Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. 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URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). 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Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. 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Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory 23, 337–343 (1977). [15] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. 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Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. 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Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. 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[58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) 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Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. 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Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  15. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [16] Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity. Entropy 20, 605 (2018). [17] Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. 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[47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Shang, M. M. & Tegnér, J. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. 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Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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[59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  17. Algorithmic complexity and reprogrammability of chemical structure networks. Parallel Processing Letters 28 (2018). [18] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). 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(eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  18. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). [19] Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Adams, A. & Abrahão, F. S. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) 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[50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) 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Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. 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URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  19. Optimal spatial deconvolution and message reconstruction from a large generative model of models. arXiv:2303.16045v1 [cs.IT] (2023). [20] Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N., Zea, A. & Tegnér, J. Causal deconvolution by algorithmic generative models. Nature Machine Intelligence 1, 58–66 (2019). [21] Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. 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URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. & Tegnér, J. A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2022). [22] Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) 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Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ming Li, Xin Chen, Xin Li, Bin Ma & Vitanyi, P. Clustering by compression. IEEE International Symposium on Information Theory, 2003. Proceedings 261–261 (2003). [23] Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. 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A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. 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[50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. 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Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. 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URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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[59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). 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Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  23. Dauwels, J. et al. Slowing and loss of complexity in alzheimer’s eeg: two sides of the same coin? International journal of Alzheimer’s disease 2011 (2011). [24] Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  24. Zenil, H. A review of methods for estimating algorithmic complexity: options, challenges, and new directions. Entropy 22, 612 (2020). [25] Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lempel, A. & Ziv, J. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  25. On the complexity of finite sequences. IEEE Transactions on information theory 22, 75–81 (1976). [26] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems 1 edn (Cambridge University Press, 2023). URL https://www.cambridge.org/core/product/identifier/9781108596619/type/book. [27] Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. 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Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Mowshowitz, A. & Dehmer, M. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. 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Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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[59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. 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  27. Entropy and the complexity of graphs revisited. Entropy 14, 559–570 (2012). [28] Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ivanciuc, O. Chemical Graphs, Molecular Matrices and Topological Indices in Chemoinformatics and Quantitative Structure-Activity Relationships§. Current Computer Aided-Drug Design 9, 153–163 (2013). [29] Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? 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Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). 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URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Von Korff, M. & Sander, T. Molecular Complexity Calculated by Fractal Dimension. Scientific Reports 9, 967 (2019). [30] Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Böttcher, T. From Molecules to Life: Quantifying the Complexity of Chemical and Biological Systems in the Universe. Journal of Molecular Evolution 86, 1–10 (2018). [31] Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. 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[37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. 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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. 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Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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[59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. 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Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
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URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Fridman, L. & Cronin, L. Lee cronin: Controversial nature paper on evolution of life and universe — lex fridman podcast #404. Youtube (2023). URL https://www.youtube.com/watch?v=CGiDqhSdLHk. [32] Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. 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Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). 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[38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. 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[51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Wolfram, S. A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [33] Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. 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[36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Delahaye, J.-P. & Zenil, H. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). 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Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. 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Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  33. Numerical evaluation of the complexity of short strings: A glance into the innermost structure of algorithmic randomness. Applied Mathematics and Computation 219, 63–77 (2012). [34] Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Krakauer, D. C. & Plotkin, J. B. Redundancy, antiredundancy, and the robustness of genomes. Proceedings of the National Academy of Sciences 99, 1405–1409 (2002). URL https://www.pnas.org/doi/abs/10.1073/pnas.032668599. [35] Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. 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[39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. 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[47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Murray, A. R. & Cronin, L. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). 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[50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) 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Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. 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URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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[59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). 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Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  35. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, 20160342 (2017). [36] Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). 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[50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) 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Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. 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URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. 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[59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). 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Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  36. Algorithmic Information Dynamics: A Computational Approach to Causality with Applications to Living Systems (Cambridge University Press, 2023). [37] Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A., Abrahão, F. S. & Tegnér, J. N. Algorithmic Information Dynamics. Scholarpedia 15, 53143 (2020). Revision #200719. [38] Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). 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Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S. & Zenil, H. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. 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The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  38. Emergence and algorithmic information dynamics of systems and observers. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2022). [39] Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Delahaye, J.-P. & Gaucherel, C. Image characterization and classification by physical complexity. Complexity 17, 26–42 (2012). URL https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.20388. [40] Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. 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Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. 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Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Kiani, N. A. & Tegnér, J. Quantifying loss of information in network-based dimensionality reduction techniques. Journal of Complex Networks 4, 342–362 (2015). URL https://doi.org/10.1093/comnet/cnv025. [41] Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. 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[44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kiani, N. A., Zenil, H., Olczak, J. & Tegnér, J. Evaluating network inference methods in terms of their ability to preserve the topology and complexity of genetic networks. Seminars in Cell & Developmental Biology 51, 44–52 (2016). URL https://www.sciencedirect.com/science/article/pii/S108495211630012X. Information Theory in Systems Biology Xenopus as a model system for vertebrate development. [42] Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). 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Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. 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Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. 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[49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. et al. Minimal algorithmic information loss methods for dimension reduction, feature selection and network sparsification (2023). 1802.05843. [43] Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. Two-dimensional kolmogorov complexity and an empirical validation of the coding theorem method by compressibility. PeerJ Computer Science 1, e23 (2015). [44] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. 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Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. 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Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. 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The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
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[51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H., Soler-Toscano, F., Jean-Paul, D. & Gauvrit, N. 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The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Algorithmic information distortions in node-aligned and node-unaligned multidimensional networks. Entropy 23 (2021). [45] Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. 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URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Abrahão, F. S., Wehmuth, K., Zenil, H. & Ziviani, A. Benito, R. M. et al. (eds) An Algorithmic Information Distortion in Multidimensional Networks. (eds Benito, R. M. et al.) Complex Networks & Their Applications IX, Vol. 944 of Studies in Computational Intelligence, 520–531 (Springer International Publishing, Cham, 2021). [46] Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. 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A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. 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[61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. 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Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. 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Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. 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The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
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URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. 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Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. & Minary, P. Training-free measures based on algorithmic probability identify high nucleosome occupancy in dna sequences. Nucleic Acids Research 47, e129–e129 (2019). URL https://doi.org/10.1093/nar/gkz750. [47] Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. 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URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Hernández-Orozco, S., Hernández-Quiroz, F. & Zenil, H. The limits of decidable states on open-ended evolution and emergence. ALIFE 2023: Ghost in the Machine: Proceedings of the 2023 Artificial Life Conference ALIFE 2016, the Fifteenth International Conference on the Synthesis and Simulation of Living Systems, 200–207 (2016). URL https://doi.org/10.1162/978-0-262-33936-0-ch039. [48] Kirchherr, W., Li, M. & Vitányi, P. The Miraculous Universal Distribution. The Mathematical Intelligencer 19, 7–15 (1997). [49] Festival, W. S. The Limits of Understanding. YouTube (2015). URL https://www.youtube.com/watch?v=DfY-DRsE86s&t=5392s. [50] Zenil, H. 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URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? 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URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. 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Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? 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Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023).
  50. Zenil, H. Turing patterns with Turing machines: emergence and low-level structure formation. Natural Computing 12, 291–30 (2013). [51] Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Marshall, S. M., Moore, D., Murray, A. R., Walker, S. I. & Cronin, L. Quantifying the pathways to life using assembly spaces (2019). URL http://arxiv.org/abs/1907.04649. [52] Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Kieffer, J. & En-Hui Yang. Grammar-based codes: a new class of universal lossless source codes. IEEE Transactions on Information Theory 46, 737–754 (2000). URL http://ieeexplore.ieee.org/document/841160/. [53] Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Lehman, E. & Shelat, A. Approximation algorithms for grammar-based compression. Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms 205–212 (2002). [54] Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Rytter, W. Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D. (eds) Grammar compression, lz-encodings, and string algorithms with implicit input. (eds Díaz, J., Karhumäki, J., Lepistö, A. & Sannella, D.) Automata, Languages and Programming, 15–27 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004). [55] Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Gańczorz, M. Entropy bounds for grammar compression. arXiv preprint arXiv:1804.08547 (2018). [56] Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Charikar, M. et al. The Smallest Grammar Problem. IEEE Transactions on Information Theory 51, 2554–2576 (2005). URL http://ieeexplore.ieee.org/document/1459058/. [57] Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Cover, T. M. & Thomas, J. A. Elements of Information Theory (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005). [58] Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. The Annals of Mathematical Statistics 32, 594–601 (1961). URL http://www.jstor.org/stable/2237768. [61] Gillen, C., Jeancolas, C., McMahon, S. & Vickers, P. The Call for a New Definition of Biosignature. Astrobiology 23, 1228–1237 (2023). [62] Malaterre, C. et al. Is There Such a Thing as a Biosignature? Astrobiology 23, 1213–1227 (2023). Ziv, J. & Lempel, A. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory 24, 530–536 (1978). [59] Welch, T. A. A technique for high-performance data compression. Computer 17, 8–19 (1984). [60] Billingsley, P. On the coding theorem for the noiseless channel. 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Citations (6)
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Summary

  • The paper demonstrates that Assembly Theory’s assembly index is mathematically equivalent to LZ compression methods in estimating algorithmic complexity.
  • It rigorously compares the performance of AT with traditional compression techniques, showing similar results in distinguishing organic from non-organic molecules.
  • The study challenges AT's claims by proving it does not capture the complex dynamics of natural selection and evolutionary processes.

An Examination of Assembly Theory Relative to Algorithmic Complexity

The work titled "Assembly Theory is an approximation to algorithmic complexity based on LZ compression that does not explain selection or evolution" scrutinizes Assembly Theory (AT), positioning it within the framework of algorithmic complexity and demonstrating its equivalence to existing compression algorithms. The authors, Felipe S. Abrahão et al., contribute a critical analysis of AT by grounding it in the context of algorithmic information theory (AIT), directly refuting claims made by AT's proponents regarding its purported novelty and explanatory power.

The paper systematically proves that Assembly Theory and its central constructs, such as the assembly index, are extensions of well-established concepts in information theory, namely lossless compression and algorithmic complexity. This is achieved through rigorous mathematical treatment and comparative analysis. The proof consists of demonstrating the equivalency of the assembly index to compression methods, specifically the Lempel-Ziv (LZ) family of algorithms, which are statistical in nature and have been used since the late 1970s.

Key Findings and Numerical Insights

The authors establish a formal equivalence between the assembly index used in AT and the method of context-free grammars, underscoring that the assembly index effectively measures complexity in a manner equivalent to calculating the size of such grammars. This equivalence reveals that the assembly index corresponds to a form of LZ compression, which is a subset of algorithmic complexity and closely tied to Shannon Entropy.

The theoretical analysis highlights that any putative discriminatory power of AT is likely due to its underlying similarity to LZ compression, which approximates algorithmic complexity by exploiting repeated patterns in data. To further assert their point, the authors use experimental data which shows that the assembly index performs equivalently or inferiorly to traditional compression algorithms in distinguishing organic from non-organic molecules.

Implications for Selection and Evolution

Critically, the paper contests AT's claim to explain natural selection and evolution beyond what is achieved by current models. By centering AT within the domain of algorithmic information theory, the authors argue it fails to account for the intricacies of biological processes that extend beyond statistical measures of complexity and compressibility. The authors posit that AT's inability to surpass existing informational and algorithmic models in capturing selection and evolution phenomena undermines its purported explanatory power.

Theoretical and Practical Developments

The implications of this study are twofold: theoretically, it reaffirms the robustness of algorithmic complexity and its adaptability across various domains, including biology and chemistry; practically, it advises a cautious scrutiny of new frameworks that repurpose established concepts without substantial novelty or empirical validation beyond existing methods.

For future research in AI and information theory, this work suggests that rather than introducing entirely new metrics, enhancing existing measures with an enriched understanding of their foundational theories may yield more substantial advances. The ongoing exploration of algorithmic complexity in causal analysis and its applications to diverse scientific questions continues to hold potential for innovation, particularly in the realms of biological and artificial systems.

In sum, the dissection of Assembly Theory within this paper provides seasoned researchers with clear insights into the intersections of compression algorithms and theories of complexity, emphasizing the necessity for empirical validation and theoretical rigor when advancing new scientific measures.

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